Answer:
0.0778 = 7.78% probability that the average age at death of these nine participants will exceed 68 years
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Approximately normal with mean equal to 66.3 years and standard deviation equal to 3.6 years.
This means that ![\mu = 66.3, \sigma = 3.6](https://tex.z-dn.net/?f=%5Cmu%20%3D%2066.3%2C%20%5Csigma%20%3D%203.6)
Sample of 9:
This means that ![n = 9, s = \frac{3.6}{\sqrt{9}} = 1.2](https://tex.z-dn.net/?f=n%20%3D%209%2C%20s%20%3D%20%5Cfrac%7B3.6%7D%7B%5Csqrt%7B9%7D%7D%20%3D%201.2)
What is the probability that the average age at death of these nine participants will exceed 68 years?
This is 1 subtracted by the pvalue of Z when X = 68. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{68 - 66.3}{1.2}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B68%20-%2066.3%7D%7B1.2%7D)
![Z = 1.42](https://tex.z-dn.net/?f=Z%20%3D%201.42)
has a pvalue of 0.9222
1 - 0.9222 = 0.0778
0.0778 = 7.78% probability that the average age at death of these nine participants will exceed 68 years